MEN OF MATHEMATICS discuss the n-valued function y which is defined, as a function of #, by the equation P»Wyn 4- PitoiT1 -f •• • + PR-i(*)ir + P»<«) = 0, in which the P's are polynomials in %. This equation defines y as an n-valued function of x. As in the case of z/2 — as = 0, there will be certain values of x for which two or more of these n values of y are equal. These values of x are the so-called branch points of the n-valued function defined by the equation. All this is now extended to functions of complex variables, and the function w (also its integral) as defined by P0(s)»" -r P^)-"-1 -r .. - + P»-i(=)w + PB(c) = 0, in which z denotes the complex variable s -f it, where s91 are real variables and i = V — 1. The n values of w are called the branches of the function aj. Here we must refer (chapter on Gauss) to what was said about the representation of uniform functions of z. Let the variable z (= s -j- if) trace out any path in its plane, and let the uniform function/(z) be expressed in the form 17 -f iV, where U, V are functions of s, t. Then, to every value of z will correspond one, and only one, value for each of U,V, and, as z traces out its path in the ss i-plane, / (z) will trace out a corresponding path in the Z7, F-plane: the path of / (2) will be uniquely determined by that of 2. But if w is a multiform (many-valued) function of z> such that precisely n distinct values of w are determined by each value of z (except at branch points, where several values of w may be equal), then it is obvious that one u^plane no longer suffices (if n is greater than 1) to represent the path, the 'march' of the function 10. In the case of a too-valued function to, such as that determined by a;2 = z> two zt'-planes would be required and, quite generally, for an n-valued function (n finite or infinite), precisely n such zo- planes would be required. The advantages of considering uniform (one-valued) func- tions instead of n-valued functions (n greater than 1) should be obvious even to a non-mathematician. What Riemann did was this: instead of the «»distinct re-planes, he introduced an n- sheeted surface, of the sort roughly described in what follows, 544